Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $z \neq 0$. $r = \dfrac{30z + 18}{-3} \div \dfrac{25z + 15}{7z} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $r = \dfrac{30z + 18}{-3} \times \dfrac{7z}{25z + 15} $ When multiplying fractions, we multiply the numerators and the denominators. $r = \dfrac{ (30z + 18) \times 7z } { -3 \times (25z + 15) } $ $ r = \dfrac {7z \times 6(5z + 3)} {-3 \times 5(5z + 3)} $ $ r = \dfrac{42z(5z + 3)}{-15(5z + 3)} $ We can cancel the $5z + 3$ so long as $5z + 3 \neq 0$ Therefore $z \neq -\dfrac{3}{5}$ $r = \dfrac{42z \cancel{(5z + 3})}{-15 \cancel{(5z + 3)}} = -\dfrac{42z}{15} = -\dfrac{14z}{5} $